Thursday, February 28, 2013

The Road Not Taken (Logic Puzzle)

In Robert
Frost’s famous poem, The Road Not Taken (sometimes mistakenly referred
to as “The Road Less Travelled”), a traveler is faced with the difficult choice
of which road he should follow.

This brings
to mind a classic logic puzzle, with mathematical implications.

Here is my
poetic interpretation of the puzzle.

THE ROAD
NOT TAKEN – Mr. Wagneezy Version

(Line 1 by
Robert Frost)

Two roads diverge in a yellow wood

One leads to certain death

The other leads to riches untold

I stop to catch my breath

I soon discover I have no clue

Exactly which road is which

I look to the right, and then to the left

My eyes begin to twitch

Suddenly two gnomes appear

From out of nearby briars

One of them is a truthful gnome

The other one is a liar

In looking at these gnomes, alas

I cannot tell the difference

Which one speaks truth?Which one speaks lies?

I fight the urge to wince

These seemingly identical gnomes

Both know which road to take

But instead of making it clear to me

They make it nearly opaque

The gnomes agree that one of them

Will answer a single question

Once I get the answer

They will end the conversation

I still can’t tell which one speaks truth

And which one is the liar

They smirk at me, these pesky gnomes

That came out of the briars

I must determine what to say

It is a daunting task

To get the gnomes to reveal the way

What question should I ask?

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Successfully
getting these gnomes to show us which road to choose will take careful
planning.Since we have no way of
knowing which gnome tells the truth and which gnome lies, we must be able to
come up with a question that both gnomes would answer the same way.

Obviously,
the direct approach (e.g. “Which road leads to untold riches?”) will not work,
because the truth-teller would point to one road while the liar would point to
the other road.

Therefore,
we must ask an indirect question – one that incorporates both the truth and the
lie.We can accomplish this by asking
one gnome to tell us which road the other
gnome would point us towards.(E.g.
“If I asked the other gnome which road leads to untold riches, which road would
he point to?”)

The logic
here is that the truth about a lie gives the same result as a lie about the
truth.

More
specifically, if we happen to talk to the truth-teller, he will point to the
wrong road because that is the road the liar would have pointed to.

On the other
hand, if we happen to talk to the liar, he will also point to the wrong road
because that is not the road the
truth-teller would have pointed to.

In either
case, the wrong road will be indicated, and we can choose the other road to
travel on.

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For movie
buffs – a version of this puzzle appeared in the movie Labyrinth
:

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Earlier, I
mentioned that this puzzle has mathematical implications.

Let’s
deconstruct this puzzle into a similar mathematical question.

I hope you
agree with me that telling the truth is positive,
and lying is negative.

Consider
mathematical operations that can be performed on two numbers.Suppose one of the numbers is positive, and
the other number is negative, but we DON’T KNOW WHICH IS WHICH.

Which
operations are guaranteed to give us results with the same sign, regardless of
which number is positive?

If we
arbitrarily choose ±2 and ±3 for our numbers and use them to explore each
operation, we get the following:

As we can
see, multiplication and division are the only operations that
fit the bill.

(Not
coincidentally, multiplication and division have the same priority in the Order of Operations.)

Therefore,
the logical argument

“The truth about a lie is equivalent to a lie about the
truth”

seems to
match up with the mathematical concepts

“A negative times a positive is equivalent to a positive
times a negative”

and

“A negative divided by a positive is equivalent to a
positive divided by a negative”.

By the way –
multiplication and division both exhibit the desired property because

1)Multiplying is the same thing as dividing by the
reciprocal.

2)Dividing is the same thing as multiplying by the
reciprocal.

and

3)A
number’s sign (positive or negative) is the same as the sign of the number’s
reciprocal.